We wish to bring a correction to Chapman et al. (2009) to the attention of readers. In the original paper, we used mixed-effects models with the effect statistic Zr (Fisher transformation of correlation coefficients) as the response, which can be written as (supposing that z denotes the n by 1 vector of effect sizes, Zr):

  • image
  • image

where β is the a by 1 vector of fixed effects, X is an n by a design matrix associated with the effect sizes, u is the b by 1 vector of random effects (we had the four random factors in our analysis: Class, Family, Species and Study) and the ith element of u is normally distributed with the mean of zero and the variance of inline image, Z is an n by b design matrix associated with the effect sizes, and e is the n by 1 vector of residual errors, normally distributed with the mean of zero and with the variance of inline image divided by (n−3); note that 1/(n−3) represents the effect-size specific measurement error variance of Zr. We inadvertently estimated inline image from the data in the original analysis. In meta-analysis, however, inline image should be fixed to 1 (inline image = 1). Fortunately, our estimates of inline image in all the original models were very close to 1 (such similarity is probably expected, because the variance of the z distribution is 1). Therefore, our original results are quantitatively very similar to those of correct meta-analytic models and our qualitative conclusions remain unchanged. We wish to refer readers to Nakagawa et al. (2011) for more detailed explanations of how correct meta-analysis can be implemented using existing computer programs. We apologize for the error.


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  2. References
  • Chapman JR, Nakagawa S, Coltman DW, Slate J, Sheldon BC (2009) A quantitative review of heterozygosity-fitness correlations in animal populations. Molecular Ecology, 18, 27462765.
  • Nakagawa S, Ockendon N, Gillespie DOS, Hatchwell BJ, Burke T (2011) Corrections for “Assessing the function of house sparrows’ bib size using a flexible meta-analysis method [Behav Ecol 18:831–840]”. Behavioral Ecology, 22, 445446.